j. finan. intermediation - olin business schoolapps.olin.wustl.edu › faculty › thakor ›...

J. Finan. Intermediation 25 (2016) 1–29

Contents lists available at ScienceDirect

J. Finan. Intermediation

journal homepage: www.elsevier .com/locate/ jfi

The highs and the lows: A theory of credit riskassessment and pricing through the business cycle

http://dx.doi.org/10.1016/j.jfi.2015.06.0031042-9573/� 2015 Elsevier Inc. All rights reserved.

Anjan V. ThakorECGI and John E. Simon Professor of Finance, Olin School of Business, Washington University in St. Louis, United States

a r t i c l e i n f o

Article history:Received 21 February 2014Available online 20 June 2015

Keywords:Financial crisisLearningSkillLuck

a b s t r a c t

This paper develops a theory of how risk is assessed and pricedthrough the business cycle by developing an intuitive model inwhich there is uncertainty about whether outcomes depend onthe risk-management skills of banks or are just based on luck, inthe spirit of Piketty’s (1995) model of ‘‘left-wing’’ and‘‘right-wing’’ dynasties. Periods of sustained banking profitabilitycause all agents to rationally elevate their estimates of bankers’skills, despite the uncertainty about what is driving outcomes.Everybody consequently becomes sanguine about bank risk, creditspreads decline, and banks choose increasingly risky assets.Conditional on subsequently observing unexpectedly high defaults,beliefs can endogenously shift to put more weight on outcomesbeing luck-driven, and this causes credit spreads to widen andmay be enough to trigger a crisis. Regulatory implications of theanalysis are extracted.

� 2015 Elsevier Inc. All rights reserved.

1. Introduction

1.1. Motivation

An interesting stylized fact about banking is how the evaluation and pricing of risk by bankers andother market participants changes over the business cycle. For example, Krainer (2004) documentsthat credit spreads shrink during expansions and widen during economic contractions/recessions.There are various reasons why spreads, and even equity risk premia, vary over the business cycle.

http://crossmark.crossref.org/dialog/?doi=10.1016/j.jfi.2015.06.003&domain=pdf

http://dx.doi.org/10.1016/j.jfi.2015.06.003

http://dx.doi.org/10.1016/j.jfi.2015.06.003

http://www.sciencedirect.com/science/journal/10429573

http://www.elsevier.com/locate/jfi

2 A.V. Thakor / J. Finan. Intermediation 25 (2016) 1–29

For example, Bekaert et al. (2013) provide evidence that this is driven at least in part by changes in riskaversion that are correlated with monetary policy changes. Moreover, monetary policy and othershocks can work through the ‘‘balance-sheet channel’’ to change the distribution of wealth that cancause a variation in spreads in general equilibrium.1 Nevertheless, there is substantial empirical evi-dence that lending standards vary over the business cycle, and that this is also a cause of variation inspreads. Asea and Blomberg (2014) examine quarterly data on two million commercial loans over the1977–1993 time period and document systematic patterns in lending standards. They find that lendingstandards decline—credit terms become easier for borrowers—during economic expansions and tightenin recessions. Similarly, Bassett et al. (2014) also document that lending standards started becomingmore lax in the U.S. in late 2003 and this easing of standards continued until early 2006, with tighteningthat started in early 2007. Further evidence is provided by Dell’Ariccia et al. (2012) who show that arelaxation of credit standards, triggered by an increased demand for loans, contributed to the boomand the ensuing financial crisis of 2007–2009.

The phenomenon is not just limited to U.S. banking. Gizycki’s (2001) empirical analysis shows thatAustralian banks weakened their credit standards during a rapid expansion of aggregate credit, whichtypically goes hand in hand with economic booms; Jiménez et al. (2012) provide similar evidence forSpain. And this collective optimism about credit risk during expansions is not just limited to bankers.As Brunnermeier (2009) notes, credit rating agencies appeared to have overly optimistic forecastsabout the creditworthiness of structured finance products (prior to the 2007–2009 financial crisis)at the same time that lending standards were becoming more lax and the market was awash in cheapcredit. Moreover, the U.S. government’s Financial Crisis Inquiry Commission (FCIC) report blamesregulators—specifically the Federal Reserve—of being too supportive of industry growth objectivesand not being vigilant enough in ‘‘reining in’’ risk taking.

Many have noted that these lax credit standards during expansions sow the seeds of future crisesas loans extended on easier terms come back to haunt banks when there is a downturn (see, for exam-ple, Asea and Blomberg (2014) and Acharya and Naqvi (2012)). During these contractions, the pricingof risk changes quite dramatically, credit standards become more stringent, and risky lending iseschewed. Bassett et al. (forthcoming) view such a tightening of credit standards as a shock to creditsupply and document a significant decline in lending as a result.

1.2. Research question

Why does the evaluation and pricing of risk change like this over the business cycle and what doesthis teach us about the risk management practices of bankers and portfolio managers? The main goalof the paper is to develop a theory to address this question. The theory is aimed at understanding sud-den reassessments of risk in a world in which rational agents are engaged in Bayesian learning, with‘‘rations optimism’’ prevailing when things are going well, and a precipitous (seemingly non-Bayesian)decline in this optimism when some ‘‘bad news’’ arrives.

1.3. The analysis

The basic version of the two-period model used to develop the theory has three key buildingblocks. The first is that banks can choose between a relatively safe loan and a potentially moreprofitable risky loan, with public observability of the type of loan chosen. The probability of success(repayment) of the loan depends on the realization of a macroeconomic state: there is a high proba-bility of a ‘‘skill’’ macroeconomic state in which outcomes are influenced by the a priori unknown skillsof banks and a small probability of a ‘‘luck’’ macroeconomic state in which these outcomes are purelyexogenous.2 No one knows at the outset which state governs the economy, but there are common priorbeliefs about the probabilities of the luck and skill states. If outcomes are driven only by luck, then banks

1 See Bernanke and Gertler (1995).2 An alternative interpretation is that there are two systematic risk regimes: high risk and low risk. In the high risk regime,

default risk is deemed to be so high that investors will fund only low-risk loans even if banks are viewed as being highly skilled,whereas in the low-risk regime, riskier loans are funded if the bank is viewed as being skilled enough.

A.V. Thakor / J. Finan. Intermediation 25 (2016) 1–29 3

prefer to invest in (and can raise financing for) only safe loans. If outcomes are skill-driven, banks willprefer to and be able to fund more profitable risky loans if these banks are viewed as being skilledenough. This building block generates a setting in which banks initially invest in safe loans, and afterexperiencing successful repayment on these loans they switch to more risky loans, conditional on agentsbelieving that outcomes depend on bankers’ skills. That is, initial success leads to an upward revision inthe skills of bankers via rational learning and these higher skill assessments enable riskier lending. Thereis symmetric information at all dates about whether outcomes are luck-driven or skill-driven and howskilled the bank is if outcomes are skill-driven.

The second building block is that there are multiple time periods and the realization of the macrouncertainty can change from one period to the next. That is, conditional on initial success with loanrepayment and agents believing that the success was skill-driven, banks are able to invest in riskierloans. However, if beliefs about whether outcomes are skill-driven or luck-driven remain intertempo-rally static, then investors will always continue to fund successful banks and risk-taking will keeprising with no financial crisis. It is only when beliefs can switch from a skill regime to a luck regimethat previously-sanctioned risk taking can fall out of favor with investors, so this possibility of regimeswitching is an essential aspect of sudden changes in risk assessments.

While adding this second building block is necessary, it is not enough to generate a drop in lendingassociated with a change in risk assessments. The reason is that if agents shift their beliefs aboutwhether outcomes are skill-driven or luck-driven, then banks will also switch at that point to loansthat investors will support given the new beliefs.3 By making their loan portfolio choices movelock-step with the beliefs of their financiers, banks can avoid a reduction in lending.

This is why we need the third building block – interim refinancing risk. For each time period, it isendogenously shown that the bank will fund itself with debt that is of shorter maturity than its loans.The reason for this is the important contribution of Calomiris and Kahn (1991) which rationalizeddemand deposits as a source of discipline on bankers. Withdrawal risk for the bank then arises fromthe regime shift pertaining to the macro uncertainty occurring before the loans for that period mature.Because each bank faces withdrawal risk only due to the realization of a systematic-risk variable, theinability of one bank to refinance itself is accompanied by a similar experience for all banks that haveinvested in that asset, so this risk reassessment actually generates a crisis.

In the basic model, there are only two time periods and two types of loans. When the model isextended to have multiple periods and multiple levels of loan risk, in order to accommodate morerealism, we get the result that initially banks invest in safe loans and if these are successfully repaid,they switch to loans of moderate risk. The presence of banks in the market for risky products createsliquidity that invites other institutions to enter, which further enhances liquidity. In this analysis, weencounter a somewhat surprising result: if repayments are experienced on these moderately riskyloans for long enough, banks switch to the riskiest loans available, bypassing all available levels of riskbetween the moderately-risky and most-risky loans. Thus, we get a form of ‘‘three-asset separation’’ interms of the bank’s asset choice—when there is a sufficiently long sequence of good outcomes, risktaking rises in steps, going from safe loans to moderately risky loans and then to the riskiest loans.

The analysis assumes that there is a time-invariant probability that outcomes are skill-determinedin every period, and that the realization of this macro uncertainty in any period tells investors whetheroutcomes are skill-based or luck-based. That is, outcomes could be viewed as being skill-based in oneperiod and luck-based in the next. But this leaves unanswered the important question of how beliefsare determined. That is, following say a long sequence of realizations that outcomes are skill-based,what could cause a belief realization that they are luck-based? To address this question, an extensionof the model is developed in which there is a true underlying reality and it is either that outcomes areskill-based or that they are luck-based. However, no one knows what the reality is, and all agentsshare common prior beliefs about the probability that outcomes are skill-based. Over time, there isupdating of beliefs about the probability distribution of the macro uncertainty based on the numberof banks that experienced successful repayments on their loans and the number of banks that failed.

3 Because of the symmetric information assumption, banks always fund the loans most profitable to their shareholders thatinvestors are willing to finance.


Thus, there are two types of learning now. One is about the macro uncertainty, which is based onobserving aggregate bank failures. The other is about the skill of an individual bank, which is basedsolely on observing the loan repayment or default outcome related to that bank. Two cohorts/gener-ations of banks – whose lending is not synchronized in time – are modeled, with the fortunes of bothcohorts being affected by the same macro uncertainty. The analysis shows that unexpectedly largedefaults in one cohort – which may be no more than the statistical realization of a low-probabilityevent – can cause beliefs about the macro uncertainty to change sufficiently to cause interim with-drawals and liquidations of banks in the other cohort.4

This modeling approach, wherein there is a regime in which people believe outcomes areskill-dependent and another in which they believe they are just luck, is reminiscent of Piketty’s(1995) model of social mobility and redistributive politics. In his model, there are two groups ofagents – those in ‘‘left-wing dynasties’’ who believe outcomes are exogenous (luck) and those in‘‘right-wing dynasties’’ who believe they depend on individual effort. The left-wing dynasties supporthigher redistributive taxation and supply less effort, while right-wing dynasties support lowerredistribution and work harder. Instead of two groups of agents, in the model in this paper thereare two states/regimes, which could be thought of as regimes distinguished by different types ofagents representing the predominant majority.5

It is important to note the causal chain of events predicted by this theory. It is when agents believeoutcomes are skill-dependent and there is a sufficiently long experience of good outcomes that riskycredit spreads shrink, risky lending is elevated and there is an economic boom.6 That is, risk-taking willtend to be ‘‘underpriced’’ by all concerned during economic booms. Moreover, since all banks are ex anteidentical in the model, when there is a shift in beliefs that outcomes are due to luck, it takes down allbanks. This should not be taken too literally. In a setting with cross-sectional heterogeneity, say onein which banks have different capital levels, for example, we would expect the higher-capital banks tobe more likely to survive.7

1.4. Related literature

This paper is related to previous theories of how banks change credit standards during the businesscycle. Rajan (1994) and Ruckes (2004) develop theories in which banks soften their lending standardsin response to competitive pressures during booms. Acharya and Naqvi (2012) build a model in whichexcess liquidity in banks leads to excessive credit volume because loan officers are incented to growloan volume. This excess lending during the good times then sows the seeds of a crisis in the future.Unlike these papers, in the model in this paper there is no excess supply of liquidity or competitivepressure that induces banks to engage in riskier lending during booms. Rather, it is a reassessmentof risk based on rational learning about the skills of bankers to manage these risks. This learning essen-tially changes credit spreads, making riskier loans cheaper to finance when banks have experienced along string of successes (no defaults). Moreover, in the model developed here, it is not just banks butall agents—including regulators and financiers of banks—who become more sanguine about risk asbanks experience success.

While the main goal of the analysis is to explain how sudden shifts in risk assessments can occurand the implications of these for risk has implications for financial crises. In the version of the modelwith more than two periods, a financial crisis occurs when banks are making the riskiest loans and

4 This analysis is an illustration of one possible mechanism by which agents’ beliefs about the macro uncertainty can evolve, andhow, in conjunction with altered beliefs about bankers’ skills, this can trigger a significant widening of credit spreads and a dryingup of funding. There may be other aggregate mechanisms as well that cause beliefs to shift. In the case of the subprime crisis,unexpectedly large defaults on subprime mortgages may have been a significant contributor to a shift in beliefs about the extent towhich bankers’ skills could influence the default characteristics of such mortgages.

5 For example, the Pew Research Center surveys citizens in different countries about whether success is due to hard work orluck. The Economist (March 9–15, 2013) reports significant differences across countries in terms of the percentage of respondentsopining that outcomes are due to luck. Thus, there may be both cross-sectional and intertemporal variations in agents’ beliefsabout what determines economic outcomes.

6 Thus, the model says nothing about whether the skills of bankers matter more in expansions or in recessions.7 Berger and Bouwman (2013) provide empirical evidence that supports this view.


market participants realize that outcomes depend only on luck. In other words, credit risk is the high-est when people think it is the lowest. This is reminiscent of Minsky (1975) who argued that optimismincreases during economic expansions, driving up speculation and leverage, thereby making the finan-cial system more fragile. While Minsky (1975) based his argument on irrationality, the theory devel-oped here shows that this can happen even with rationality when there is a regime-switchingpossibility. Thus, this paper can be viewed as Minsky (1975) meeting Piketty (1995) in a financial crisissetting. This aspect of this paper connects it to the vast literature on financial crises.

Historical accounts of financial crises that have occurred over the years appear in Bordo et al.(2001), Gorton (1988), Kindleberger (1978), and Reinhart and Rogoff (2008). The real consequencesof financial crises are documented in Atkinson et al. (2013), Boyd et al. (2005), Kupiec and Ramirez(2013), and Laeven and Valencia (2012). Bernanke (1983) discusses the real effects of the banking cri-sis during the Great Depression. Wilms et al. (2014) provide a review of the real effects of crises.

Papers that review the crisis literature include Allen and Gale (2007), Lo (2012), Rochet (2008), andThakor (forthcoming). The events surrounding the 2007–2009 subprime crisis are described in greatdetail in Brunnermeier (2009) and Gorton (2010), among others. Many papers have been written onthe causes of financial crises (e.g. Acharya and Yorulmazer (2008), De Jonghe (2010), and Wagner(2010)). Some have pointed to political forces (e.g. Johnson and Kwak (2010), Rajan (2010), andStiglitz (2010)),8 whereas monetary policy in the U.S. and Europe has been viewed as a major contributorby others (e.g. Maddaloni and Peydro’ (2011) and Taylor (2009)).9 Yet others have pointed to incentiveconflicts of various sorts (e.g. Kane, 1990, forthcoming; Farhi and Tirole, 2012). More recently, complex-ity, interconnectedness and innovation have appeared in theories of financial crises and contagion.Interconnectedness and complexity are modeled in Allen et al. (2012) and Caballero and Simsek(2013). Gennaioli et al. (2012) point to ‘‘neglected risks’’ in innovative financial products, and Thakor(2012) models the role of disagreement over the value of innovative financial products as causal factorsin crises. The neglected risk modeled in Gennaioli et al. (2012) is a behavioral bias. For other behavioraltheories, see Gennaioli et al. (2015) and Thakor (2015).10 In both these papers, agents irrationally over-react to good news, which leads to excessively risky lending. Finally, the view that seems to be gatheringthe most support is that the combination of excessive leverage and maturity mismatching by financialinstitutions have contributed significantly to financial crises. For theories along these lines, seeAcharya and Thakor (2014), Shleifer and Vishny (2010) and Farhi and Tirole (2012). Excessive leveragealso plays a role in the theory developed by Goel et al. (2014), but in that theory, it is the correlated natureof leverage on banks’ and households’ balance sheets that creates fragility and makes the system vulner-able to a crisis. They also provide supporting evidence.

In contrast to these papers, the focus in this paper is not on incentive conflicts, leverage, complexityor innovation. The central idea in this paper, and one that has not been previously examined as a cau-sal factor in sudden reassessments of risk that sometimes lead to financial crises, is thatexperience-based learning, whereby outcomes are attributed primarily to skill but have a non-zeroprobability of being viewed in the future as being just luck, can create an environment for banks totake successively higher levels of risk that eventually leads to a crisis. Leverage-financed lendingbooms followed by a sudden contraction of liquidity due to a sharply higher (seeminglynon-Bayesian) spike in perceptions of risk are features common to many crises. This paper explainsthat such sharp revisions in assessments of risk can occur even with Bayesian rational agents.

I do not mean to suggest that learning is the only factor that—operating through bank risk manage-ment—affects the likelihood of banking crises. The impact of government-provided safety nets—likedeposit insurance—on risk taking cannot be overlooked; see Thakor (2014) for a discussion of theseissues in the context of the impact of bank capital on financial stability. Demirgiic-Kunt and

8 Rajan (2010) points out that politicians were interested in expanding home ownership in the U.S., and therefore strengthenedthe Community Reinvestment Act (CRA). Agarwal et al. (2012) provide evidence that this encouraged banks to pursue riskiermortgage lending.

9 The argument here is that easy-money policies adopted by central banks in the U.S. and Europe contributed to housing pricebubbles in various countries.

10 Thakor (2015) develops a regime-switching model, but one in which the availability heuristic bias causes agents to ignore thelikelihood that outcomes could be due to luck. This leads to excessive exuberance during periods of sustained economic booms.


Detragiache (1999) study 61 countries and document that explicit deposit insurance seems to bedetrimental to bank stability. Calomiris and Haber (2014) attribute this to the moral hazard createdby safety nets with a growing tendency to shift the losses of bank failures to taxpayers, and they sug-gest that banking fragility reflects the structure of a country’s fundamental political institutions. Astriking aspect of this that Calomiris and Haber (2014) highlight is that some countries have hadnumerous banking crises and others (Canada, New Zealand, Australia, Singapore, Malta and HongKong) have had none, a puzzle typically not addressed in theories of banking crises. As an illustrationof this, they compare the U.S.—which has had 12 major banking crises since 1840—with Canada, whichhas had none. They point to a number of factors, with political economy roots, to explain this. In par-ticular, they point out that the Canadian banking system had no branching restrictions and no depositinsurance, factors that led to consolidation, diversification and a focus on prudent risk managementand solvency among Canadian banks. These arguments suggest that we should think of the interactionbetween learning and political economy generating differences in risk management across bankingsystems in different countries. Supporting this view is the emphasis in Calomiris and Haber (2014)on the differences in underwriting standards across U.S. and Canadian banks prior to the 2007–2009 crisis in the U.S.; underwriting standards declined significantly at U.S. banks, but not atCanadian banks.11 In other words, if bank structure and regulation provide strong incentives for banksto avoid risky lending, then the impact of learning in whetting the risk appetites of banks and investors isretarded.

The rest is organized as follows. The two-period (five dates) model is developed in Section 2, andanalyzed in Section 3. An analysis of the bank’s loan portfolio and funding choices with more than twoperiods and more than two types of loan appears in Section 4, and it shows that a sufficiently pro-longed sequence of good outcomes eventually leads to the bank financing the riskiest loans available.The analysis of how beliefs about the macroeconomic uncertainty can change based on the observa-tion of aggregate defaults appears in Section 5. Section 6 is devoted to a discussion of the implicationsof the analysis for financial crises and regulatory policy, and the empirical prediction of the model.Section 7 concludes. All proofs are in the Appendix A.

2. The model

This section describes the base model with five dates and two time periods. I begin with a descrip-tion of agents and preferences. I then discuss the sequence of events, followed by the investmentopportunity set. This is followed by a statement of the observability assumptions and description ofhow beliefs are revised. I conclude with a summary of the timeline and a discussion of why the variouselements of the model are needed.

2.1. Preferences of key players

Everybody is risk neutral and the riskless rate is zero. The key players in the base model are finan-cial intermediaries and institutional investors who can provide financing for these intermediaries. Theliabilities of the intermediaries are uninsured,12 so even though I will refer to these intermediaries as‘‘banks’’ henceforth, it should be understood that they encompass a broad array of financial intermedi-aries that raise their funding via non-traded and other claims in the capital market, including investmentbanks and commercial banks. For commercial banks, this would be deposit funding (including uninsureddeposits), and for investment banks it could be traded debt, repos, as well as longer-term borrowing fromother institutions. Thus, the financial market here includes the shadow banking system.

11 Calomiris and Haber (2014), p. 324 write: ‘‘Canada’s banking system avoided the 2007–2009 banking crisis that crippled theUnited States and much of Western Europe. The relaxation of underwriting standards that we discuss in Chapter 7 produced aquintupling of residential mortgage arrears in the United States. In contrast, Canadian residential mortgage arrears displayedcharacteristic, and enviable, dullness.’’ Their characterization of lax underwriting standards prior to the crisis at U.S. and Europeanbanks is consistent with the theory developed here.

12 The model goes through with partially insured liabilities.


2.2. Sequence of events: agents, information signals and types of loans

There are five payoff-relevant dates: t = 0, 1, 2, 3 and 4 that cover two time periods over whichbanks operate. At t = 0, there are N0 banks in the market. Each bank can choose to invest $1 at t = 0in either a prudent loan (P) or a risky loan (R). Both loans mature at t = 2. For simplicity, the entire$1 is raised at t = 0 in the form of (uninsured) debt financing through non-traded debt supplied byinstitutional investors; the bank’s choice between such debt and direct capital market financing willbe considered later. The debt is competitively priced to yield investors a zero expected return. Debtmaturity will be endogenized.

2.3. Bank manager and loan monitoring

The bank’s manager needs to monitor the loan in order to produce a positive payoff. If the loan isnot monitored at t = 0, it produces a payoff of 0 at t = 2, regardless of whether it is a P or an R loan. Thesame is true for the second-period loan. If not monitored at t = 2, it produces a zero payoff at t = 4. Thebank manager’s utility for each period can be written as

13 Forrelation

Uði; jÞ ¼W þ a ½value of bank equity with loan i and monitoring decision j� � Cjþ BIfsurviveg

ð1Þ

where W is the manager’s fixed up-front wage each period, a is a positive constant, j 2 {1, 0} wherej = 1 if the bank monitors and j = 0 if the bank does not monitor, C > 0 is the manager’s personal costof monitoring, B is the personal benefit to the manager from the bank surviving until the end of theperiod, and I{survive} is the indicator function which is 1 if the bank survives until the end of the periodand zero otherwise. It will be assumed that:

B > C ð2Þ

so that the manager’s personal benefit from survival exceeds his personal monitoring cost.The debt investors who finance the bank receive a signal at the interim date (t = 1) for the

first-period loan and t = 3 for the second-period loan that informs them whether the bank monitoredits loan or not. If the loan was not monitored, then its value is Ln 2 (0, 1) if it is liquidated at the interimdate (t = 1 or t = 3), where Ln is an arbitrarily small positive quantity. Since the value of an unmoni-tored loan is zero at the end of the period, debt investors will wish to liquidate the loan at the interimdate if they discover it was not monitored.

2.4. Investment opportunities: P and R loans

The bank chooses at t = 0 and then again at t = 2 between loans P and R, a mutually-exclusive set ofrelationship loans.13

2.4.1. P loansLoan P is either good (G) or bad (B) but no one can determine for sure a priori whether the loan is G

or B. Conditional on being monitored, a loan of type G pays off Xp > 1 with probability (w.p.) 1, and a Bloan pays off Xp w.p. b 2 (0, 1) and 0 w.p. 1 � b.

It is commonly believed at t = 0 that there is a state of nature, m, in any given period, which couldbe interpreted as a macroeconomic state. The realization of this state at date t, denoted, mt, has twopossible values: ‘‘skill’’ and ‘‘luck’’. In the ‘‘luck’’ state (probability k 2 (0,1)), outcomes are pure luck,so the probability that the loan is type G is fixed at r 2 (0, 1) for all banks. In the ‘‘skill’’ state (proba-bility 1 � k), outcomes are skill-dependent, so the probability that the loan is type G depends on theskill/talent of the bank in monitoring the loan after it is made, and more talented banks are viewed ashaving higher probabilities of making good loans. Specifically, there are two possible types of banks:

a theory of relationship lending, see Boot and Thakor (1994, 2000). Relationship lending allows the banks to earnship-related rents, as in the model here.


talented (s) and untalented (u). A type-s bank monitors the P loan with perfect efficiency and is thusable to ensure that the loan is G w.p. 1. A type-u bank, however, has no monitoring ability and thusends up with a B loan w.p. 1.14 The common prior belief at t = 0 is that the probability that any givenbank is type-s is r, and the probability that it is type u is 1 � r. The banks themselves share these beliefs,so there is no private information about the bank’s type.

In any given period, the realization of the skill or luck state applies to P loans made by all banks, butacross the two periods, these state variables are identically and independently distributed (i.i.d.) ran-dom variables. That is, the realization of mt does not provide any new information about the likelihoodof a particular realization of mt+1.15

The macro state – skill or luck – is independently and identically distributed (i.i.d.) across time peri-ods, and all agents share the common prior belief that the probability of the luck state in any period isk. In the base model, it is assumed that, at the interim date (t = 1 for the first period and t = 3 for thesecond period), which state has been realized for the period can be costlessly discovered by all, includ-ing the bank’s creditors. Later, this information will be permitted to be discovered only at a cost.

While one may interpret the skill and luck states as representing two different objective physicalregimes, they may also represent nothing more than subjective assessments of agents about whethereconomic outcomes are driven by skill/effort or just luck. That is, the skill and luck macro states can beinterpreted as states of ‘‘aggregate investor sentiment’’, in the spirit of Piketty’s (1995) model, with theluck state being one in which market sentiment is dominated by agents in Picketty’s ‘‘left-wing dynas-ties’’, and the skill state being one in which market sentiment is dominated by agents in Picketty’s‘‘right-wing dynasties’’. A shift from one state to the other may be triggered by events in the real econ-omy or changes in political initiatives/rhetoric. For example, higher-than-expected defaults on sub-prime mortgages may cause investors to shift from beliefs that loan repayment outcomes aredetermined by the screening skills of bankers to the belief that they are driven largely by the bad luckexperienced by subprime borrowers. Alternatively, there may be an objective underlying reality, as inthe extended version of the model in Section 5.

At t = 0 then, the prior belief about the probability of success of the P loan at t = 2 is

14 Thiw.p. les

15 If thbecomesame stpresentperiodsoutcom

rPo � r þ ½1� r�b ð3Þ

It is assumed that

rPoXp > 1 ð4Þ

so P can be financed with debt given these prior beliefs.However,

bXp < 1 ð5Þ

so a bank known to be type-u almost surely (in the state in which the probability of having a type-Gloan is skill-dependent) would never be able to raise financing for a P loan.

2.4.2. R loans

The R loan can be either good ðbGÞ or bad ðbBÞ. No one can determine a priori whether a given loan is bGor bB. A loan of type bG pays off XR w.p. q 2 (0, 1) and 0 w.p. (1 � q), whereas a bB loan pays off 0 w.p. 1. It is

assumed that the bG-type R loan has a higher expected value to financiers than the G-type P loan, i.e.,

C þ aa

> q½XR þ Z� > XP þ Z ð6Þ

s assumption is merely to reduce notational clutter, and one could assume instead that a type-u bank ends up with a B loans than 1.e macro uncertainty exhibited positive serial correlation, the results will get even stronger, although the algebra willmore messy. A macro uncertainty realization of skill in one period will indicate a higher (than prior) probability of the

ate in the second period, making elevated risk-taking even more attractive. In Section 5, a variant of the model will beed in which there is only one underlying reality (either outcomes are luck-based or they are skill-based) across all time, and economic agents learn over time, based on observing aggregate defaults, updating their beliefs about whetheres are based on luck or skill.


where Z > 0 is a non-pledgeable payoff that is available to the bank’s shareholders if the loan stays onthe books until it matures, is repaid and is not prematurely liquidated.16

In (6), the first inequality, Cþaa > q½XR þ Z�, means that the manager’s equity ownership is never suf-

ficient to induce him to monitor the loan, in the absence of a liquidation threat by creditors. The sec-ond inequality, q[XR + Z] > XP + Z, means that the R loan has a higher expected value to financiers thanthe P loan.

As in the case of the P loan, there are two possible macroeconomic states of relevance with the R loan

in any given period. In the ‘‘luck’’ state (which has probability (k), the loan is type bG with an exoge-

nously fixed probability r. In the ‘‘skill’’ state (w.p. 1 � k), the probability that the loan is type bG isdependent on the bank’s monitoring skill. In this case, the common belief is that a type-s bank will

be able to ensure w.p. 1 that it is a bG loan, whereas a type-u bank will ensure w.p. 1 that it is a bB loan.It will also be assumed, for later use, that the difference in loan repayment (success) probabilities

across the good and bad loans is greater for the R loan than for the P loan, i.e.,

16 Thecreditorthat arewhichcreatesshareho

Table 1Loan pa

q > 1� b ð7Þ

This is motivated by the observation that R is a more complex loan for which the importance of thebank’s skill/talent is greater than for P.

At t = 0 then, the prior probability of success of the type R loan at t = 2 is:

rRo � rq ð8Þ

and it is assumed that

rRo ½XR þ Z� < 1 ð9Þ

so the R loan cannot be financed with debt or equity given the prior beliefs.A summary of these loans and payoffs is provided in the table below.

principal role of Z is to create a wedge between the interests of the bank’s creditors and those of its shareholders. Thus,s may wish to liquidate the bank even when shareholders would like to continue, which permits a focus on liquidationsgenerated by the short-term nature of debt funding rather than voluntary bank closures by its shareholders. Instances in

the interests of shareholders and bondholders diverge ex post are not hard to find. For example, if continuation merelya mean-preserving spread of the payoff distribution with liquidation, creditors will prefer liquidation even thoughlders prefer continuation.

yoffs.


2.5. Observability and knowledge assumptions

Each bank’s loan choice at any given date is commonly observable. So, investors know the bank’schoice (P or R) before providing financing. Also publicly observable is whether the bank’s loan repaidor defaulted at the end of the period (t = 2 for the first period and t = 4 for the second period). For now,mt (skill or luck) is observable to all once it is realized. There are common beliefs about each bank’stype at each date t, so no bank knows more about its type than others do.

2.6. Non-pledgeable payoffs

The payoff Z is non-pledgeable in the sense that it cannot be pledged by the bank as repayment tocreditors. Each loan also has an additional social value J P 0; that is available only if the loan is suc-cessfully repaid. This social value is another non-pledgeable payoff. It can be interpreted as a benefit tosociety from the success of the borrower’s project due to its positive employment, consumer welfareand tax-revenue-enhancing wealth generation consequences.

2.7. Financing from institutional investors

Institutional investors provide financing in a competitive private debt market, so that bank debt ispriced to yield investors an expected return of zero. If investors liquidate a (monitored) loan at t = 1 ort = 3, they collect L 2 (rqXR, rq[XR + Z]), where rq[XR + Z] < 1. The assumption that L < rq[XR + Z] meansthat a bank liquidation is inefficient ex post.

2.8. Definition of a financial crisis

A financial crisis occurs if all banks in the system are prematurely liquidated at either t = 1 or t = 3.

2.9. Summary of timeline

At t = 0, there are N0 banks. Each bank chooses to invest $1 in either a prudent (P) loan or a risky (R)loan. Each loan is entirely funded by uninsured debt and each matures at either t = 1 or t = 2.17 Thedebt will be called ‘‘short maturity’’ if it matures at t = 1 and ‘‘long maturity’’ if it matures at t = 2.18

At t = 1, based on their discovery of the realized value of m1, investors who funded the bank at t = 0may wish to collect their repayment at t = 1 or renew funding until t = 2. In the event that funding isnot renewed, the bank must either repay investors in full or liquidate. If the bank survives until t = 2without liquidation, then it collects its loan payment (if available) and pays off investors. Any positiveprofit at t = 2 on the first-period loan is paid out as a dividend to the bank’s shareholders.19 The lendingcycle then restarts at t = 2 and the bank makes a new choice of loan and funds it entirely with new debt.This debt is ‘‘short maturity’’ if it matures at t = 3 and ‘‘long maturity’’ if it matures at t = 4. The ‘‘skill’’ or‘‘luck’’ macro state realization for the second period, m3, occurs at t = 3, and this is observed at that timeby all in the second period. The terminal payoff is realized at t = 4.

A summary of the sequence of events is provided in Fig. 1. In Fig. 2, there is a pictorial summary ofthe probability distributions of the projects.

2.10. Discussion of model

The model has numerous moving parts, so it is useful briefly review the role these elements play inthe model and why they are needed.

17 It makes little difference to the analysis if the bank was partly financed with equity, as long as it is not predominantly financedwith equity. Non-trivial amounts of debt financing are essential for fragility.

18 It will be shown in the analysis that the short maturity of the bank’s debt emerges endogenously.19 This is for simplicity. If the bank could use its first-period profit to fund part of its second-period loan, it would reduce the

reliance on outside debt. However, the analysis is qualitatively unchanged if the bank has to fund the loan even partly from outsidedebt.

0 1 2 3 4

• Banks choose between prudent (P)and risky (R) loans and invest $1 in the chosen loan. The bank’s choice of Por R is publicly observed.

• The loan is entirely debt financed, and financing is raised from institutional investors or through traded debt.

• The common prior belief is that there is a probability 1 λ−that the loan repayment probability is skill-dependent, and in this case the probability is

(0,1)r∈ that the

bank is talented (type τ ) and 1-rthat it is untalented (type-u). There is a probability λ of the macro state

1 luckm = (the loan repayment probability is purely exogenous).

• Information about 1m is

costlessly available to

investors.

• Investors observe 1m as

well as whether bank

has monitored first-

period borrower and

decide whether to

liquidate the bank or

renew funding.

• If the bank can pay off

investors their promised

amount by raising new

debt, it survives until

t=2. Otherwise, it may

cease to exist.

• Loans made at t=0

pay off and

investors are paid

off if bank not

liquidated at t=1.

• Investors revise

their beliefs about

the bank’s type

based on whether

the loans pay off or

not.

• Banks make new loans financed with new debt.

• Investors learn 3m

and whether the bank

monitored its second-

period borrower.

• Investors then decide

whether to liquidate

the bank or renew

funding.

• Loans made at t=2

pay off and new

investors paid off

if bank not

liquidated at t=3.

Fig. 1. Sequence of events.


2.10.1. Two periods – five datesTwo periods are the minimum needed to accommodate learning about banks on the part of both

banks and their investors, so that funding renewals can be made contingent on observed outcomes.The five dates are needed to allow for interim withdrawal risk in each period in addition to the loangranting and repayment dates in each period.

2.10.2. Bank monitoring of loans at a private cost to bankerThis is needed to endogenize maturity mismatching by the bank in a manner similar to the market

discipline argument in Calomiris and Kahn (1991). While it is analytically most convenient to modelthis as post-lending monitoring in the Holmstrom and Tirole (1997) sense, one can also interpret it aspre-lending screening in the Ramakrishnan and Thakor (1984) sense. This means that the loans in thismodel can be viewed either as business loans where post-lending monitoring matters a great deal, ormortgages and other kinds of consumer loans where screening is more important.20

20 For empirical evidence on the importance of screening for mortgage lending, see Keys et al. (2010) and Purnanandam (2011).

Fig. 2. Pictorial depiction of project success probabilities.


2.10.3. P and R loansThese two types of loans are needed to give the bank an asset portfolio choice. The potential

dependence of the loan payoffs on the bank’s skill is at the heart of the learning model.

2.10.4. Macro state uncertaintyThis uncertainty is the core of the analysis because it permits discontinuous changes in risk

assessments with Bayesian rational agents.

3. Analysis of the base model

The main goal of this section is to analyze the model developed in the previous section to examinehow banks fund themselves, their debt maturity and their loan portfolio choices.

3.1. Bank debt maturity

In this subsection, the question of the bank’s debt maturity is examined. The following parametricrestrictions will be imposed on the model.

W > C > afqXR � 1g ð10Þ


the first part of the restriction merely says that the manager’s up-front wage is enough to cover hisloan monitoring cost, which is sufficient for his participation constraint to be satisfied. The second partof the restriction says that, even if the bank was identified unambiguously as type s and it invested inloan R, the net expected payoff to the manager (a fraction a of the payoff to the bank’s shareholders) isexceeded by his personal cost of monitoring. This creates a moral hazard that the bank’s creditorsmust account for. The following result is now immediate.

Proposition 1. In each period the bank will finance its loan with short-maturity debt. That is, in the firstperiod, it will issue debt at t = 0 that matures at t = 1 and has to be replaced with short-maturity debt thatmatures at t = 2. In the second period, it will issue short-maturity debt at t = 2 that matures at t = 3 and isreplaced with short-maturity debt that matures at t = 4.

The intuition is straightforward. In the absence of any refinancing risk – which would be the casewith long-term debt – the bank manager would not monitor. This would make it impossible for thebank to raise financing. Thus, short-term debt is the bank’s only viable financing alternative.

3.2. Bank loan market: analysis of outcomes at t = 2

In the usual backward induction manner, I start by analyzing the second period, beginning at t = 2,first and then the first period beginning at t = 0.

3.2.1. Default/repayment on first-period loanSuppose the bank made a P loan at t = 0. There are now two possible states for the bank at t = 2; (i)

its loan defaults at t = 2, or (ii) its loan pays off.

State (i): Repayment failure (P loan defaults at t = 2): In this case, the common posterior belief aboutthe bank’s type at t = 2 is:

s f2 ðskillÞ ¼ Prðsjdefault on first-period loan; m1 ¼ skillÞ ¼ 0 ð11Þ

if it was observed that m1 = ‘‘skill’’. And it is

s f2 ðluckÞ ¼ Prðsjdefault on first period loan; m1 ¼ luckÞ ¼ r ð12Þ

if it was observed that m1 = ‘‘luck’’. We can thus compute the expected success probabilities on the

second-period projects, conditional on having observed first-period failure. Using notation Pij and Ri

j,j 2 fS;Dg; i 2 f1;2g to designate the event j of either successful repayment (S) or default (D) in periodi (which is 1 or 2, corresponding to repayment outcomes observed at t = 2 for period 1 and t = 4 forperiod 2) on loans P and R respectively, one can write the posterior probabilities at t = 2 as:

PrðP2S jP

1D;m1 ¼ luckÞ ¼ PrðP2

S jP1D;m1 ¼ luck;m3 ¼ luckÞPrðm3 ¼ luckÞ

þ PrðP2S jP

1D;m1 ¼ luck;m3 ¼ skillÞPrðm3 ¼ skillÞ

conditional on m1 = luck having been observed. Using (12) yields:

PrðP2S jP

1D; m1 ¼ luckÞ ¼ krP

0 þ ½1� k�rP0 ¼ rP

0 ¼ r þ ½1� r�b ð13Þ

Similarly, using (11):

PrðP2S jP

1D; m1 ¼ skillÞ ¼ krP

0 þ ½1� k�b ð14Þ

Note that @PrðP2S jP

1D; m1 ¼ skillÞ=@k > 0. Thus, for k small enough, it will be true that

PrðP2S jP

1D; m1 ¼ skillÞ½XP þ Z� < 1 ð15Þ

which means that, subsequent to first-period default and the knowledge that m1 = skill, no financingwill be available for the P loan in the second period. It will be assumed throughout that (15) holds.

Similarly, using (11) and (12):


PrðR2S jP

1D; m1 ¼ luckÞ ¼ krR

0 þ ½1� k�rR0 ¼ rR

0 ¼ rq ð16Þ

PrðR2S jP

1D; m1 ¼ skillÞ ¼ krR

0 þ ½1� k�½0� ¼ krR0 < rR

0 ¼ rq ð17Þ

where k and 1 � k above refer to the macro state probabilities in the second period.Since an R loan cannot be funded using prior beliefs (see (9)), it cannot be funded following

first-period default. This gives us Lemma 1.

Lemma 1. A bank that makes a P loan in the first period and experiences default will exit the market ifm1 = skill and will make a P loan in the second period if m1 = luck.

State (ii): Repayment success (P loan repaid at t = 2): In this case, the common posterior belief aboutthe bank’s type at t = 2 is:

sS2 ðskillÞ ¼ PrðsjP1

S ; m1 ¼ skillÞ ¼ rr þ b½1� r� > r ð18Þ

if it is observed that the ‘‘skill’’ state occurred in the first period. And it is

sS2ðluckÞ ¼ PrðsjP1

S ;m1 ¼ luckÞ ¼ r ð19Þ

if it is observed that the ‘‘luck’’ state occurred in the first period. Now, one can write

PrðP2S jP

1S ;m1 ¼ luckÞ ¼ rP

0 ð20Þ

PrðP2S jP

1S ;m1 ¼ skillÞ ¼ krP

0 þ ½1� k�½sS2ðskillÞ þ ½1� sS

2ðskillÞ�b� ð21Þ

where sS2 ðskillÞ is given by (18). Similarly,

PrðR2S jP

1S ;m1 ¼ luckÞ ¼ rR

0 ð22Þ

PrðR2S jP

1S ;m1 ¼ skillÞ ¼ krR

0 þ ½1� k�qsS2ðskillÞ ¼ krqþ ½1� k�qsS

2ðskillÞ ð23Þ

Now, conditional on repayment of the P loan in the first period and m1 = skill, the total value of a Ploan in the second period is:

PrðP2S jP

1S ;m1 ¼ skillÞ½XP þ Z þ J� ð24Þ

which includes its social value. Similarly, the total value of an R loan is

kLþ ½1� k�qsS2 ðskillÞ½XR þ Z þ J� ð25Þ

if it is assumed that, conditional on m3 = luck, the bank’s creditors will liquidate it at t = 3 if it investedin an R loan in the second period.

3.2.2. Second-period lending choiceNow consider the second-period lending choice for a bank whose first-period P loan was repaid,

and for which m1 = skill. If it makes a P loan at t = 2, its expected profit is:

�pP2 ¼ PrðP2

S jP1S ;m1 ¼ skillÞfXP � DPðPrðP2

S jP1S ;m1 ¼ skillÞÞ þ Zg ð26Þ

where DP is the promised repayment on the debt financing raised by the bank at t = 2 to finance itsloan.

The bank’s repayment obligation to debt investors should be such that the amount raised fromthese investors at t = 0 ($1) equals the expected payoff to them at t = 2, i.e.

DPðPrðP2S jP

1S ;m1 ¼ skillÞ ¼ 1=PrðP2

S jP1S ;m1 ¼ skillÞÞ ð27Þ


I now turn to the R loan. If we assume that the bank will be liquidated in the second-period if itinvests in the R loan at t = 2 and m3 = luck at t = 3, then the bank’s expected profit from an R loan is:

21 In tloans if

22 Thilikeliho

�pR2 ¼ ½1� k�qsS

2 ðskillÞfXR � DRðPrðR2S jP

1S ;m1 ¼ skillÞÞ þ Zg ð28Þ

where the bank’s repayment obligation is:

DRðPrðR2S jP

1S ;m1 ¼ skillÞÞ ¼ ½1� kL�f½1� k�qsS

2g�1 ð29Þ

The following result can now be stated.

Lemma 2. For k 2 ð0;1Þ and b 2 (0, 1) sufficiently small and q 2 (0, 1) sufficiently large, we have:

½1� k�qsS2 ðskillÞXR > PrðP2

S jP1S ; m1 ¼ skillÞXP þ ½PrðP2

S jP1S ;m1 ¼ skillÞ � ½1� k�qsS

2ðskillÞ�Z ð30Þ

The implication of (30) is that a bank that experienced repayment success on the first-period P loanand observed m1 = skill will have such a high posterior probability of success on the R loan in the sec-ond period that the bank’s shareholders’ expected payoff will be higher with an R loan than with a Ploan. It will be assumed throughout that the conditions stated in Lemma 2 will be satisfied, so (30)holds.

The following result is now easy to derive.

Proposition 2 (Bank’s Second-Period Lending). If its first-period P loan repaid at t = 2 and m1 = skill, thebank prefers to invest in the R loan in the second period at t = 2. It raises $1 in financing at t = 2 to invest inthe second-period loan at t = 2 and promises creditors a repayment of L if investors collect on their credit at

t = 3, and DRðPrðR2S jP

1S ;m1 ¼ skillÞÞ if they provide renewed funding at t = 3 that requires repayment

at t = 4. If the bank’s creditors verify at t = 3 that the bank monitored its loan, then they will renew fundingat t = 3 if m3 = skill and liquidate the bank if m3 = luck.

The intuition is as follows. Based on the previous analysis, if the bank’s first-period loan success-fully repays, then the posterior belief about the bank’s probability of making a second-period loan thatwill repay is so high that it becomes profitable for the bank to make an R loan and financiers will fundit. If there is no renewal of the loan at t = 3, the bank has to liquidate, so it collects L to pay creditors.Conditional on the bank having monitored its loan, the creditors will wish to do this only if m3 = luck,because in this case beliefs about repayment probabilities revert to prior beliefs. If m3 = skill, thencreditors get a higher expected payoff if they allow the bank to continue than if they liquidate it.

Since all banks are identical in this model, all those who experience success on their first-period Ploan will invest in R loans in the second period. This is essentially the whole banking industry sincebanks that experienced first-period default have exited at t = 2. The exit of a bank that experiencesdefaults is an outcome of the parametric restrictions imposed, and the assumption that all loans ofa particular type have perfectly correlated prospects.21 More generally, if a bank makes multiple loansof the same type (P or R) with imperfectly correlated prospects, then whether a bank that experiences(some) first-period defaults exits or not will depend both on the number of defaults and the bank’s lever-age. Banks with fewer defaults and higher capital levels may survive,22 but what will be true is that theywill stick to safe lending and make only P loans in the second period, regardless of the realization of m1.Hence, the broader risk-management implication of Proposition 2 is that banks that experiencefirst-period loan repayment success pursue riskier lending in the second period, whereas banks thatexperience (some) loan defaults stick to safe lending or are forced to exit.

Strictly within the model, since only banks that experienced (complete) loan repayment at t = 2 areable to continue, and these banks invest in R loans in the second period, an observation of m3 = luck att = 3 leads to a sharp reassessment of risk. We can interpret this as a discontinuous increase in the

he model, each bank makes only one loan, but this is the same as the bank making any arbitrarily large number (N > 1) ofthe prospects of all loans of the same type are perfectly correlated.

s is consistent with the empirical evidence in Berger and Bouwman (2013) that banks with higher capital have a higherod of surviving a financial crisis.


credit spread or risk premium on R loans, which leads to liquidations of all banks and a financial crisis.Note also that Lemma 2, which states the conditions under which banks switch to the R loan, requiresthat k 2 ð0; 1Þ be sufficiently low. Since a crisis occurs only when the luck state is realized, this alsoimplies that the ex ante probability of a crisis must be low enough for a crisis to occur ex post. Ifthe probability of the luck state is sufficiently high, banks will never choose the R loan and investorswill never fund it.

3.3. Analysis of outcomes at t = 0

The following result is now immediate, given our earlier analysis.

Lemma 3 (Banks First-Period Lending). At t = 0, all banks invest in P loans. Investors are promised arepayment of L at t = 1 and DPðrP

0Þ at t = 2, where

DPðrP0Þ ¼ ½rP

0��1 ð31Þ

and the bank’s shareholders’ expected profit is:

pP1 ¼ rP

0½XP � DPðrP0Þ þ Z� ð32Þ

Investors do not liquidate the bank and agree to renew financing at t = 1 regardless of the realized value ofm1. There is no reassessment of risk in the first period.

The reason there is no reassessment of risk in the first period is that the realized value of m1 doesnot cause investors to change their beliefs from their prior beliefs about the bank’s ability since norepayment/default on any loan has been observed. The reason why the bank invest in P is that R can-not be financed with investors’ prior beliefs about the bank’s ability.

We see then that banks start out investing in P loans, and then those that experience successfulrepayment go on to make R loans in the second period, as long as the probability of m3 = luck is smallenough (see Lemma 2). But if m3 = luck is indeed observed at t = 3, there is a sharp upward reassess-ment of risk. This reassessment leads to liquidations of banks, which are ex post inefficient bothbecause of the non-pledgeable rents, Z, that these banks’ shareholders lose, but also due to the lossof additional social rents, J. These liquidations affect all banks, so it is a crisis. The following resultis now useful to note.

Lemma 4 (Social Welfare). If the non-pledgeable social rent, J, is large enough, then the bank may invest inR in the second period even though it is socially inefficient compared to investing in P.

The intuition is that P always has a higher repayment probability than R, and hence there is a lowerprobability that the social rent J will be lost with P than with R. This means the bank’s shareholdersmay wish to invest in R and financiers will be willing to provide financing even though these loansare socially inefficient compared to the P loans. This can justify regulatory portfolio restrictions or cap-ital requirements that reduce the attractiveness of R loans for the bank. The next result establishesthat the restrictions on the deep (exogenous) parameters of the model do not lead to an empty setof permissible values.

Proposition 4. The set of exogenous parameters satisfying the restriction on the model ((2), (4), (5), (6),(7), (9), (10), (15) and (30)) is non-empty.

4. A T-period model: the effect of more time periods

Imagine now that the previous analysis represented the first two periods of an arbitrarily long timehorizon, i.e., t = T > 4 is an arbitrarily large number. Also, instead of a single R loan, there is a large setof R loans, say {R1, R2, ...,RN}, with N arbitrarily large. The probability distribution for loan Ri is that a

type-s bank will be able to ensure w.p. 1 that it is a bGi loan, where bGi loan pays off XiR w:p: q, and

0 w:p: 1� q, and XiR > X j

R if i > j, whereas a type-u bank will ensure w:p: 1 that it is a bBi loan, where


a bBi loan generates a payoff of XiR w.p. bR and a payoff of �[i � 1]k w.p. 1 � bR, where k > 0 is a positive

constant and 0 < bR < min {0.5, q} < 1. Specifically, it is assumed that XiR ¼ Xi�1

R þ ½i� 1�k8i 2 f2; . . . ;Ng.As in the previous analysis, rqXi

R < 18i. That is, it is assumed that X1R ¼ XR, where XR was defined in

Section 2 (see Table 1). For simplicity, let us continue with the assumption that the bank’s creditorscan costlessly observe the realization of m at each odd date: t = 1, 3,. . ..

A parametric restriction will be imposed on r, which says that r is small enough that

23 Onemakesbank to

r½1� r��1< ½1� 2bR�½q��1 ð33Þ

This restriction ensures that, evaluated at the prior belief that the bank is type s, riskier loans are lessattractive to the bank.

Let the liquidation value of any loan be L 2 (L1, L2), where

L1 � ½X1R þ ½N � 1�k�½rqþ ð1� rÞbR� � ½1� r�½1� bR�½N � 1�k;

L2 � minfL1 þ Z;X1R½rqþ ð1� rÞbR�g

ð34Þ

which means that all liquidations are inefficient. Note that, given (33), we know that L1 < L2.

A couple of points are worth noting. First, unlike the two-period model, the bad loan, bB, has a pos-itive probability of success. This is needed in a multi-period setting to prevent the posterior belief thatthe bank is type s, conditional on a successful outcome of a risky loan in the luck macroeconomic state,from becoming 1. Second, the bank’s payoff on the risky loan in the default state is �[i � 1]k, whichcan be viewed as the bank’s loss given default, a measure of risk that goes up with i. That is, viewedin this sense, loan Ri is riskier than loan Rj, if i > j.23

Now the probability of success with a P loan at t = 0, rp0, is given by r, and we will assume (4) and (5)

hold. The probability of success with a R1 loan at t = 0 is:

rR10 ¼ rqþ ½1� r�bR ð35Þ

The analog of (6) is

q½X1R þ Z� > Xp þ Z ð36Þ

The analog of (7) is

q� bR > 1� b ð37Þ

and the analog of (9) is

rR10 ½X

1R þ Z� < 1: ð38Þ

Now, the value of risky loan Ri, to financiers, evaluated at the prior beliefs is:

EVi ¼ rq X1R þ Z þ ½i� 1�k

h iþ ½1� r� bR X1

R þ Z þ ½i� 1�kh i

� ½1� bR�½i� 1�kn o

ð39Þ

The following result is useful.

Lemma 5. ðiÞ @EVi=@r > 0; ðiiÞ @EViðrÞ=@i < 0; and ðiiiÞ @2EVi=@i@r > 0:

The implications of this lemma are as follows. From (i), we see that as the (posterior) belief that thebank is type s increases, so does the value of any risky loan, Ri. However, (ii) implies that evaluated atthe prior belief r, riskier loans are less valuable. And, (iii) says that the marginal impact of an increasein the posterior belief that the bank’s type is s is greater for larger values of i, i.e., for riskier loans.

We now need additional notation. We know that if the bank invests in a P loan at t = 0, then theposterior probability at t = 2 that the bank is of type s, conditional on success at t = 2 and m1 = skill,is given by (18) as:

may ask what it means to have a negative payoff for the bank in the loan default state. The idea here is that the banknumerous investments of tangible and intangible capital that can only be recovered if the loan repays. Default causes the

lose these investments.


sS2 ðskillÞ ¼ PrðsjP1

S ;m1 ¼ skillÞ ¼ rr þ b½1� r�

The posterior belief at t = 2 about the probability of success in the second period for R1 (at t = 4) is:

PrðR21jP

1S Þ ¼ krR1

0 þ ½1� k�fqsS2 ðskillÞ þ ½1� sS

2 ðskillÞ�bRg ð40Þ

and, it shall be assumed that:

PrðR21jP

1S Þ½X

1R þ Z� > PrðP2

S jP1S Þ½Xp þ Z� ð41Þ

Moreover, let sS2ðskillÞ be small enough that we have the following analog of (33):

PrðR21jP

1S Þ 1� PrðR2

1jP1S Þ

h i�1< ½1� 2bR�½q��1 ð42Þ

Essentially, the restrictions imposed by (41) and (42) guarantee that the bank will switch from the P tothe R1 loan at t = 2 if its first-period P loan succeeded; note that (42) is sufficient for R1 to be preferredto any other Ri, i > 1.

Now, for an odd date t, define sSt ðskillodd

t ; Sevent Þ as the posterior probability at date t that the bank is

type s, where the vector skilloddt ¼ ðskill1; skill3; skill5; . . . skilltÞ indicates that the macroeconomic state

was revealed to be m = skill at all odd dates up to (and including) date t, and the vectorSeven

t ¼ ðS2; S4; S6; . . . St�1Þ indicates that the loans made by the bank successfully repaid at all even

dates up to (and including) date t � 1. Then, we see that sSt ðskillodd

t ; Sevent Þ is defined recursively by:

sSt ðskillodd

t ; Sevent Þ ¼ qsS

t�2

fqsSt�2 þ bR½1� sS

t�2�gð43Þ

The following result can now be proved:

Lemma 6. There exists a large enough (odd) number, t*, such that @EViðsSt Þ=@i < 08t < t� and

@EViðsSt Þ=@i > 08t P t�, where sS

t is short-hand for sSt ðskillt ; S

event Þ.

This lemma says that if there is a sufficiently long sequence of successful loan outcomes along witha correspondingly long sequence of consecutive mt = skill states, the posterior belief about the bankbeing of type s will be high enough to ensure that riskier loans will have higher expected values.This suggests that, after a long enough string of successful loan outcomes during a ‘‘bull market’’ econ-omy, the bank will switch to the riskiest loan available. This leads us to the main result of this section.

Proposition 3. In a credit market equilibrium with T (arbitrarily large) periods and costless discovery ofthe macro state m, the bank will invest in the P loan in the first period. Conditional on the first-period loanbeing successfully repaid and m1 = skill, the bank invests in the R1 loan in the second period. Conditional oneach R1 loan successfully paying off in each period and mt = skill being realized at dates t = 3, 5, . . ., the bankwill continue to invest in R1 each period until date t < t* (defined in Lemma 6). At any date t < t*, if investorsobserve that mt = luck has occurred, they will not liquidate any banks. However, if the number of periodsthat the R1 loan is successfully repaid in conjunction with mt = skill being realized is greater than or equal tot*, then risk assessments will be favorable enough that the bank will switch to investing in RN in the nextperiod and will continue to do so in the periods that follow, conditional on each loan being successfullyrepaid and mt = skill being realized. If mt = luck is realized at any t > t*, investors will liquidate all bankswith RN loans and a financial crisis will ensue.

This proposition, depicted graphically in Fig. 3, makes a number of points. First, there is a‘‘three-asset separation’’ here. The bank begins by investing in the P loan, then switches to R1, condi-tional on some events, and then continues with R1 until it switches to RN. There is no financial crisiseven if mt = luck at any time t prior to t⁄, when banks switch to RN, the riskiest loan. That is, a suffi-ciently long sequence of good outcomes can lower the risk premium on the riskiest loan enough tomake it attractive for the bank. For t > t⁄, an mt = luck realization triggers a crisis. Thus, there has tobe a sufficiently long ‘‘bull market run’’ – with the macroeconomic state being mt = skill and banksexperiencing good loan outcomes – before a crisis occurs. Prior to this, there is a very high level of

Fig. 4. Intuition for three-asset separation.

Fig. 3. Risk-taking and crisis with many time periods: three-asset separation.


risk-taking by banks that is not considered very risky by banks, investors or regulators. That is, it is notonly risk management within banks that tolerates risky loans, but also investors and regulators whodo not view these as unacceptably risky.

The intuition for this result is that riskier loans have values that are more sensitive to the bank’sskill, and thus increase more rapidly as perceptions of the bank’s skill goes up. Thus, when skill per-ceptions are relatively low, either P or R1 is the preferred loan. Since these loans have values, evaluatedat prior beliefs about bank skill, that exceed the loan liquidation values, investors do not liquidatethese loans even if mt = luck occurs. However, when skill perceptions are sufficiently high, the creditspread narrows enough that the RN loan has the highest expected value, so banks switch to it. But, ifmt = luck occurs in any period, skill perceptions drop down to the prior belief, the credit spread widensand a crisis occurs because the expected value of RN, evaluated at the prior belief, is lower than theliquidation value. This intuition is illustrated in Fig. 4.

5. How and why do beliefs about outcome determinants shift?

An interesting question is: what determines whether investors believe that outcomes areskill-dependent or luck-dependent? Thus far it has been assumed that the realization of mt 2 {luck,skill} is entirely exogenous. In this section, a slightly modified version of the model is presented inwhich there is learning about whether outcomes are determined by luck or by skill.


Consider a six-date world with t = 0, 1, 2, 3, 4, 5. There are two cohorts of banks. At t = 0, the firstcohort, consisting of NO banks, enters the market and makes loans. Let Nt be the number of banks inthis cohort at date t. Then at t = 1, the second cohort, consisting of M1 banks, enters the market andmakes loans. Let Mt be the number of banks in this cohort at date t P 1. For simplicity, no new banksenter after t = 1, and any bank that fails at any given date exits the market after that date. Cohort sizesare thus weakly decreasing over time.

Each cohort of banks starts out with the same prior beliefs about each bank’s ability and a choicebetween the P and R loans as in the base model in Section 2. Instead of assuming that in each period mt

is realized and this determines whether outcomes are skill-dependent or luck-dependent, it will nowbe assumed that there is a ‘‘true’’ state: in every period outcomes are determined either by skill or byluck, i.e., mt ¼ m 2 fluck; skillg;8t but no one knows whether it is luck or skill that determines out-comes. The common prior belief is that the probability that outcomes are determined by luck isk0 2 ð0; 1Þ and the probability that outcomes are determined by skill is 1� kO. This belief will beupdated at date t based on how many banks in each cohort experience success. A loan made att 2 {0, 1, 2, 3} matures at t + 2 unless prematurely liquidated at t + 1.

Here is the sequence of events. At t = 0, NO banks in cohort 1 make P loans since that is all investorswill fund, given prior beliefs. At t = 1, M1 banks in cohort 2 make P loans. At t = 2, NO � n2 of the NO

banks experience success and n2 experience failure and exit, so N2 = NO � n2. Based on the observationof this, all agents update their prior beliefs about the common state, so the posterior probability thatoutcomes are determined by luck is now k2. At this stage, the financiers of the cohort 2 banks candetermine whether to liquidate the loans or continue funding them. At t = 2, the surviving N2 cohort1 banks make new loans that mature at t = 4. At t = 3, M1 �m3 cohort 2 banks experience success andm3 experience failure and exit. So M3 = M1 �m3. Based on this, agents update their beliefs about thecommon state and the new posterior belief is that the probability that outcomes are determined byluck is k3. At this stage, financiers of the second set of loans made by the cohort 1 banks determinewhether to liquidate the loans or allow the bank to continue. The surviving cohort 2 banks then maketheir second round of loans. At t = 4, N2 � n4 of the N2 cohort 1 banks experience success, and n4 expe-rience failure. Based on this, a new posterior belief, k4, emerges about the common state and financiersof the cohort 2 banks decide whether to liquidate their banks or let them continue until t = 5. Themodel ends at t = 5. In this setting, note that relevant changes in beliefs about the macro state canoccur only at t = 2, 3 or 4. See Fig. 5.

Based on the previous analysis in Section 3, we know that at t = 0 and t = 1 the cohort 1 and cohort2 banks will make only P loans. Consider now what might happen at t = 2. The cohort 1 banks willinclude n2 banks that fail and NO � n2 = N2 that succeed. This outcome does not change beliefs aboutthe skills of the cohort 2 banks, and we know from the previous analysis that the failed cohort 1 banksexit, whereas the successful cohort 1 banks will invest in R loans (given the conditions in Lemma 2).

How will agents revise their beliefs and arrive at k2? This is addressed below

Lemma 7. Regardless of the number of cohort 1 banks that succeed at t = 2

k2 ¼ kO ð44Þ
It should not be surprising that beliefs about the common state are unchanged by the observed
number of failures and successes among the cohort 1 banks at t = 2. The reason is that with the P loanthe expected probability of success is the same regardless of whether outcomes are determined byskill or luck, given prior beliefs. A similar result appears below.

Lemma 8. Regardless of the number of cohort 2 banks that succeed at t = 3,

k3 ¼ kO ð45Þ
The intuition for this lemma is the same as that for Lemma 7. The cohort 2 banks invested in P loans
at t = 1. Moreover, the outcomes at t = 2 related to the loans made by cohort 1 banks reveal no infor-mation about the abilities of cohort 2 banks. Thus, k3 remains at kO. This indicates that the first updat-ing that leads to a posterior belief that differs from the prior about the common state occurs at t = 4.We now have the main result of this section.

t=0 t=1 t=2 t=3 t=4 t=5

• Cohort 1

banks make

loans in the

first round

• Cohort 2

banks make

loans in the

first round

• First round

loans of

cohort 1

banks either

pay off or

default

• Surviving

cohort 1

banks make

second round

of loans

• Investors in

cohort 2

banks determine

whether to

liquidate

cohort 2

banks or not

• First round

of loans of

cohort 2

banks pay

off or default

• Investors in

cohort 1

banks decide

whether to

liquidate

banks or

continue

• Second

round of

cohort 1

banks' loans

pay off or

default

• Investors in

cohort 2

banks decide

whether to

liquidate

these banks

or continue

• Second round

of cohort 2

banks' loans

pay off or default

Fig. 5. Sequence of events with learning about the macro state.


Proposition 5. At t = 4, if a sufficiently large number of the R loans made by cohort 1 banks fail, theposterior belief about the common state changes to k4 < kO; and this may cause the financiers of the cohort2 banks that invested in R loans at t = 3 to force the liquidation of these banks, precipitating a crisis.

The R loans made at t = 2 were made by banks that experienced success on their first-period Ploans. Thus, these banks have sufficiently high posterior beliefs about their types (conditional onm = skill), sS

2 ðskillÞ, that they could finance R loans (see (18) for sS2 ðskillÞÞ. If a very large fraction of

these R loans fail, it is in a state of the world in which a relatively small fraction of loans are expectedto fail if outcomes were truly determined by skill. Financiers infer, therefore, that it is more likely thatoutcomes are determined by luck and lower the probability they assign to outcomes being determinedby skill. This drop causes the expected probability of success on an R loan – with the expectation beingtaken over the skill and luck macro state – to decline relative to its value at t = 3 when the cohort 2banks made their R loans. If this decline is sufficiently large, then the expected success probabilitybecomes so low that financiers decide to liquidate all the R loans made by cohort 2 banks at t = 3.

This result illustrates one mechanism by which beliefs can shift across the skill and luck macrostates to dramatically alter risk assessments. In the base model, the transition in beliefs is stark andsudden. But the analysis in this section shows that it need not be so. Even a smoother belief transitioncan have the effect of substantially elevating risk assessment.

6. Welfare implications, related issues and policy implications

This section discusses a number of issues. First, I discuss the welfare implication of the model. Thisis followed by a discussion of the implications of the model for risk management in banks through the


cycle and financial crises. I end the section with a discussion of empirical predictions and regulatorypolicy implications.

6.1. Efficiency

A bank liquidation is always ex post inefficient in this model because the value of the bank, includ-ing the value of its non-pledgeable assets, is greater than the liquidation value. Nonetheless, creditorsforce liquidation because they do not have access to the non-pledgeable assets. Moreover, due to theloss of the social benefit, J, of bank continuation, liquidations and the crisis they precipitate are also exante inefficient.

6.2. Risk assessment and risk management through the cycle

The analysis shows that the longer banks do well in terms of experiencing low defaults, the morethe risk premium on risky loans relative to safe loans shrinks and the more attractive risky lendingbecomes for banks. Risk managers in banks will tend to get ‘‘marginalized’’ and credit rating agencieswill assign ratings that may seem overly optimistic with the benefit of post-crisis hindsight. Note thatthis happens due to Bayesian rational learning and does not require banks/investors to ignore (tail)risks, as in Gennaioli et al. (2012). Regulators too will sanction these risk management practices inbanks, even in the absence of regulator-taxpayer conflicts, as in Boot and Thakor (1993) and Kane(1990).

However, when banks experience defaults or beliefs shift and investors believe outcomes areluck-dependent, risk assessments change in the opposite direction as credit spreads widen, risk man-agement in banks becomes more conservative, and risky lending is eschewed. To the extent that loandefaults are higher in recessions, these risk assessments are more likely during recessions as well, sorisk taking is likely to be substantially curtailed during such times. This suggests that risk-taking islikely to be procyclical, with banks making increasingly risky loans during economic booms andbecoming more ‘‘risk averse’’ during downturns.

6.3. Financial crises

The Reinhart and Rogoff (2008) evidence suggests a systematic pattern of high leverage, corre-lated risky lending and asset price bubbles prior to financial crises. While there is no capital struc-ture decision for the bank in this model, the analysis does show that a financial crisis is precededby risky lending,24 and a high price for risky loans that looks like a price bubble when the crisisarrives.

Moreover, when we consider the triggering mechanism in Section 5, there is a similarity betweenthat mechanism and what occurred during the 2007–2009 crisis. Unexpectedly large defaults on sub-prime mortgages and escalating risks in even highly-rated securities backed by those mortgagescaused the beliefs of investors and institutions to shift and led them to conclude that the assets theythought were safe were no longer safe. The problem was exacerbated when institutions that had doneso well for so long began to experience difficulties, which caused liquidity to evaporate and theAsset-Backed Commercial Paper (ABCP) market to collapse.

Crises in other countries have followed similar patterns. For example, the financial crisis in Swedenin the early 1990s, which saw non-performing loans at Swedish banks increase to 11% of GDP in 1993,was also preceded by a substantial credit expansion and a real estate boom that saw house prices dou-bling between 1981 and 1991. Similarly, the Japanese financial crisis that started in 1991 was pre-ceded by asset price bubbles in real estate and in the stock market.

24 Since only successful banks survive and all invest in R after surviving, the pre-crisis lending decisions of banks are clearlycorrelated in our model.


6.4. Empirical predictions and policy implications

The first prediction of the analysis is that a financial crisis will occur only when all agents share thecommon belief ex ante that the probability of a crisis is very low, and when bank risk-taking is at itshighest. In particular, bank risk-taking will proceed in three steps: relatively safe lending at first, fol-lowed by loans of moderate risk, and then if there is a sufficiently long period of success, banks engagein the riskiest lending available.

This result is based on Lemma 2 and Proposition 3. The reason why a low ex ante probability ofoccurrence of a crisis is a necessary condition for a crisis to occur is that it is only when the probabilityof the luck state occurring is low enough that banks are willing to invest in very risky loans, somethingthat is a necessary precursor to a crisis. This is because it is only in that event that the elevated per-ceptions of bankers’ skills following many periods of success remain at relatively high levels, and ittakes sufficiently high ability perceptions to make very risky loans both attractive to banks and accept-able for investors to fund. As for the second part of the implication, we have extensively discussed the‘‘three-fund separation’’ result earlier.

The second prediction is that lending-related risk management in financial institutions will tend tobe procyclical, with more aggressive risk taking in economic booms and more conservative(lower-risk) lending in downturns.

The third prediction is that crises are cyclical phenomena and are (almost unavoidably) highlylikely to occur following a long period of relatively high banking profits.

I now turn to the regulatory implications of the analysis. The first implication follows readily fromthe earlier results. It is that even if regulators exhibit no agency problems vis-à-vis taxpayers, they willbe less vigilant when banks are doing well, and unlikely to prevent investments in highly-risky assets,simply because they will believe, like everyone else, that banks are skilled enough to manage the risks.

The second implication is that regulators should attempt, on an ex ante basis, to put in place mech-anisms that make risk-taking more expensive for the bank’s shareholders when the bank is doing well.One simple mechanism for doing this is higher (equity) capital requirements that are countercyclical.Recalling the discussion of the interaction of learning and political economy in the Introduction, scal-ing back on government safety nets and attending to banking industry structure will also facilitate thegoal of arresting bank risk taking.

There are examples of countries that have adopted such measures. In recent years, macroprudentialpolicies in India dynamically varied risk weights and provisioning requirements for certain classes ofreal estate loans, and Colombian rules specifically targeted loan market overheating in 2008 withhigher capital and provisioning requirements, as well as other policy changes. Countries like Canada,Australia and New Zealand, that have avoided banking crises, have all had banking systems with a rel-atively small number of large banks with significant charter values, so the cost of risk taking—the pos-sible loss of bank charter—has always been high for the bank’s shareholders, a feature that dampensrisk taking, even with learning. Of course, other policies, such as relatively high interest rates that kepta house price bubble from forming, also contributed to these countries escaping the 2007–2009 crisis.

Since in my model the probability of success of the risky loan, Ri, is always less than the probabilityof success of the prudent loan, P, higher capital requirements make Ri less attractive than P for thebank’s shareholders. By raising the capital requirement when banks are doing well—and the longerthe industry has done well, the higher the requirement should be—risk-taking is made more expensiveprecisely when it is more attractive for banks.25

In a nutshell, the analysis here provides a straightforward way to understand numerous stylizedfacts related to bank risk management through the cycle, crises, the apparent ‘‘marginalization’’ of riskmanagers in banks during periods of high banking profitability, high compensation for bank CEOs, andhigh pre-crisis liquidity in asset markets as well as seemingly lax regulatory monitoring of banks priorto crises. Moreover, the theory developed here explains why banks, regulators and investors all seemto simultaneously ‘‘underestimate’’ risk and why market participants do not adjust contracts to

25 This argument does not rely on equity capital being more costly than debt financing. It is simply based on the idea that a bankknows that a riskier investment increases the odds of losing its capital in the event of default, as in monitoring model ofHolmstrom and Tirole (1997) and the sorting model of Chan et al. (1992).


correct the apparent misalignment or increase credit spreads immediately prior to the crisis in antic-ipation of the impending consequences of such behavior. This theory suggests that many of the reg-ulatory initiatives that emerged in response to the recent financial crisis may not succeed inpreventing future crises. These initiatives include restrictions on executive compensation in banking,pricing safety nets like deposit insurance more accurately and making the pricing risk-sensitive, put-ting restrictions on regulatory forbearance (e.g. of the sort put in place after the FDICIA in 1991), align-ing the incentives of regulators more closely with those of taxpayers, engaging in more activemonitoring of systemically important financial institutions (SIFIs), etc.

7. Conclusion

This paper has proposed a theory of bank credit risk management through the business cycle. Thetheory is based on rational learning that leads to revisions in inferences of banking skills in an envi-ronment in which skills may or may not matter, depending on a macro state that could be shaped byinvestor sentiment or developments in the real sector. In such a setting, a sufficiently long sequence offavorable outcomes for banks leads all agents—banks themselves, their investors, and regulators—toassign relatively high probabilities to the abilities of banks to manage their own risks. This providesbanks with access to low-cost funding, and encourages banks to engage in riskier lending.Consequently, either if agents can directly observe that outcomes are just due to luck or observationof aggregate defaults leads to such an inference, there is a sharp increase in the risk premia on riskyassets. Under some conditions, a crisis occurs as debt investors withdraw their funding from banks.26

Future research should be directed at better understanding, in a formal way, the potential interac-tion between the learning modeled here and the political economy hypothesis of Calomiris and Haber(2014). In particular, the regulatory policy implications of this interaction should serve to illuminate avariety of vexing issues in bank regulation design and open up new questions for research.

Acknowledgments

On an earlier version of the paper (with a different title), I received helpful comments for which Ithank two anonymous referees, the editors (Charlie Calomiris and Murillo Campello), Elena Carletti,Itay Goldstein (discussant at 2012 Financial Intermediation Research Society (FIRS) meeting), JohanMaharjan, David Levine, Yiming Qian, (especially) Diego Vega and participants at the 2012 AmericanEconomic Association meeting (Chicago, January 2012), the 2012 FIRS meeting (Minneapolis, June2012), and at seminars at the European University Institute (Florence, October 2011), the Universityof Iowa (November 2011), the University of Pittsburgh (December 2011), Southern MethodistUniversity (December 2011), Erasmus University (Rotterdam, Netherlands, October 2012), theFederal Reserve Bank of Philadelphia (October 2012), the Federal Reserve Bank of Cleveland (May2013), and the Federal Reserve Bank of San Francisco (May 2013). I alone am responsible for any errors.

Appendix A

Proof of Proposition 1. Suppose investors provide the bank with a long-maturity debt at t = 0 thatmatures at t = 2. Then, given (15), the bank manager will prefer not to monitor the loan, since theprobability that he will lose his personal benefit, B, by not monitoring is zero. Hence the creditors’expected payoff at t = 2 will be zero, which means they will be unwilling to extend credit. Withshort-term debt, if creditors discover at t = 1 that no loan monitoring was done, then they willliquidate the bank and the bank manager will lose B. Given (2), it follows that he will prefer tomonitor the loan in this case. This means short-term debt is the only viable financing instrument forthe bank. h

26 While the trigger for a change in beliefs in this paper is a sufficiently high number of aggregate defaults, there may be othertriggers as well. For example, the adoption of new government policies may be another trigger.


Proof of Lemma 1. If m1 = skill and the bank experiences default on its first-period P loan, then from(15) and (17), we know that neither the P nor the R loan can be funded in the second period, so thebank must exit. If m1 = luck, then beliefs about the success probabilities of the P and R loans revertto those that existed at t = 0. h

Proof of Lemma 2. The total value of a bank to its financiers, conditional on m1 = skill, successfulrepayment of the first-period P loan and investment in the R loan in the second period is:

kLþ ½1� k�qsS2ðskillÞ½XR þ Z� ðA:1Þ

If we assume that the loan will be liquidated at t = 3 conditional on m3 = luck. If the bank chooses toinvest in a P loan in the second period, then its value to its financiers is (recalling (21)):

PrðP2S jP

1S ;m1 ¼ skillÞ½XP þ Z� ¼ ½XP þ Z�fkrP

O þ ½1� k�½sS2 ðskillÞ þ f1� sS

2ðskillÞgb�g ðA:2Þ

Label the expression in (A.1) as (A.1) and the expression in (A.2) as (A.2). It can be seen that (30) issufficient for (A.1) > (A.2). Moreover, @(A.1) / @k < 0 and @(A.1)/@q > 0. Further, we also see that@(A.2)/@b > 0. Thus, by continuity, (30) holds for k and b sufficiently small and q sufficiently large,given that (30) holds trivially when q ¼ 1; k ¼ b ¼ 0. h

Proof of Proposition 2. Suppose the bank experiences successful repayment of its first-period loan att = 2 and m1 = skill. Then, for the second-period loan choice, compare �pR

2; the expected profit from an Rloan with �pP

2, the expected profit from a P loan, where:

�pP2 ¼ PrðP2

S jP1S ;m1 ¼ skillÞfXP � DPðPrðP2

S jP1S ;m1 ¼ skillÞÞ þ Zg ðA:3Þ

where DPðPrðP2S jP

1S ;m1 ¼ skillÞÞ is by (27). Thus, substituting (27) in (A.3) yields:

�pP2 ¼ PrðP2

S jP1S ;m1 ¼ skillÞXP � 1þ PrðP2

S jP1S ;m1 ¼ skillÞZ

< PrðR2S jP

1S ;m1 ¼ skillÞ½XR þ Z� � 1 given Lemma 2:

¼ PrðR2S jP

1S ;m1 ¼ skillÞ½XR � DRðPrðR2

S jP1S ; m1 ¼ skillÞÞ þ Z�

¼ �pR2:

Thus, the bank prefers an R loan to a P loan in the second period, conditional on first-period success.Since refinancing at t = 3 is possible with probability 1, investors who provided short-term financing att = 2 that matures at t = 3 will view their claim as riskless.If creditors discover at t = 3 that m3 = skill,then they assess their expected payoff from refinancing the bank and getting repaid at t = 4 asqsS

2ðskillÞDRðPrðR2S jP

1S ;m1 ¼ skillÞ, which exceeds the payoff with liquidation at t = 3. If m3 = luck, then

investors prefer to not renew funding and force liquidation since L > rqXR. h

Proof of Lemma 3. The proof is obvious given earlier arguments. h

Proof of Lemma 4. We know that the bank will invest in R in the second period, conditional uponsuccessful repayment of P in the first period and m1 = skill. In this case its total value (including socialvalue) is:

kLþ ½1� k�qsS2 ðskillÞ½XR þ Z þ J� ðA:4Þ

If it invests in P, the total value is:

PrðP2S jP

1S ;m1 ¼ skillÞ½XP þ Z þ J� ðA:5Þ

For (A.5) to exceed (A.4), we need

J½PrðP2S jP

1S ;m1 ¼ skillÞ � ½1� k�qsS

2ðskillÞ�> kLþ ½1� k�qsS

2 ðskillÞ½XR þ Z� � PrðP2S jP

1S ; m1 ¼ skillÞ½XP þ Z� ðA:6Þ


We know that

PrðP2S jP

1S ;m1 ¼ skillÞ ¼ krP

0 þ ½1� k�½sS2ðskillÞ þ ½1� sS

2ðskillÞb� > ½1� k�sS2ðskillÞ > ½1� k�qsS

2ðskillÞ

And that the right-hand side (RHS) of (A.6) is positive (but finite) by Lemma 2. Thus, for J large enough,(A.6) will hold. h

Proof of Lemma 5. Using (39) we see:

@EVi=@r ¼ ½q� bR�½X1R þ Z þ ½i� 1�k� þ ½1� bR�½i� 1�k > 0 since q > bR ðA:7Þ

Next, using (39), we have:

@EVi=@i ¼ rqkþ ½1� r�fbRk� ½1� bR�kg ¼ kfrqþ ½1� r�bR � ½1� r�½1� bR�g < 0 given ð33Þ

Finally, using (A.7),

@2EV1=@i@r ¼ ½q� bR�kþ ½1� bR�k > 0: �

Proof of Lemma 6. Since @EVi(r)/@i < 0, we know that by continuity, 9�sSt > r such that @EVið�sS

t Þ=@i < 0:Since from (43) we know that @sS

t=@t > 0; and since @2EV1/@i@r > 0, we can assert @EVið�sSt Þ=@i < 0 for

t < t*, for some t*. Now, it follows that limt!1

sSt ¼ 1: Moreover, writing (33) as r < [1 � 2bR][1 � r][q]�1,

it can be seen that, the left-hand side converges to 1 as r ? 1, whereas limr!1½1� r�½1� 2bR�½q��1 ¼ 0.

Thus, for t* large enough, the inequality in (33) is reversed and @EViðsSt Þ=@i > 08t P t�. h

Proof of Proposition 3. The proof that the bank will invest in the P loan in the first period follows fromprevious arguments. Moreover, since investors in the bank and the bank agree on values and the inves-tors always price debt so as to break even, the previous analysis indicates that we can simply make theloan choice determination based on its total expected value. The fact that the back switches at t = 2from the P loan to the R1 loan follows from the restrictions (41) and (42) since R1 is preferred toRj8j P 2 when the probability is sS

2ðskillÞ that the bank is type s.Using Lemma 6, we know that (33) holds with sS

t ¼ r; and in that case, the economic value of R1

exceeds the value of any Rj; j P 2: When t P t�; (33) is reversed and EVi/@i > 0, so RN has the highestvalue, and the bank switches to RN. h

Proof of Proposition 4. The following exogenous parameter values satisfy all of the restrictions:

B ¼ 1:5; C ¼ 1; Z ¼ 0:02; M ¼ 0:008; r ¼ 0:6; b ¼ 0:6; W ¼ 2; a ¼ 0:3; k ¼ 0:05; XP

¼ 1:2; q ¼ 0:77; XR ¼ 2:1: �

Proof of Lemma 7. At t = 2, suppose N2 banks out of N0 are successful. Recall that the expected prob-ability of success on a P loan made at t = 0 is r + [1 � r]b. Thus,Pr (m = skill|N2 out of N1 banks experi-enced success)

¼

N0

N2

� �fr þ ½1� r�bgN2f½1� r�½1� b�gN0�N2 ½1� k0�

N0

N2

� �fr þ ½1� r�bgN2f½1� r�½1� b�gN0�N2 ½1� k0�

þN0

N2

� �fr þ ½1� r�bgN2f½1� r�½1� b�gN0�N2 k0

8>>><>>>:

9>>>=>>>;

¼ ½1� k0� �

ðA:8Þ


Proof of Lemma 8. The proof is similar to that of Lemma 7. h

Proof of Proposition 5. If N4 out of N2 banks succeed at t = 4, then we can write:

Pr ðm ¼ skilljN4 out of N2 banks succeeded at t ¼ 4Þ

¼

N2

N4

� �½sS

2ðskillÞq�N4 ½1� sS2ðskillÞq�N2�N4 ½1� k0�

N2

N4

� �½sS

2ðskillÞq�N4 ½1� sS2ðskillÞq�N2�N4 ½1� kO�

þN2

N4

� �½rq�N4 ½1� rq�N2�N4 kO

8>>><>>>:

9>>>=>>>;

ðA:9Þ

where we recognize that the probability of success of an R loan, conditional on m = skill, is sS2 ðskillÞ;

given by (18), and the probability of success of an R loan, conditional on m = luck, is rq.Now note that,ignoring the integer problem, $N4 small enough such that

Pr ðm ¼ skilljN4 out of N2 banks succeeded at t ¼ 4Þ

< 1� kO ðA:10Þ

To see this, note that the satisfaction of (A.10) requires that

fsS2 ðskillÞqgN4 ½1� sS

2 ðskillÞq�N2�N4< ½rq�N4 ½1� rq�N2�N4 ðA:11Þ

Suppose N4 = 0. Then (A.11) becomes

1 <1� rq

1� sS2 ðskillÞq

� �N2

ðA:12Þ

which clearly holds since sS2 ðskillÞ > r.Thus, it has been shown that (A.10) holds for N4 = 0. By conti-

nuity, and ignoring the integer problem, it holds for N4 small enough. Then the bank’s financiers assessthe expected payoff from continuation as

k4rqXR þ ½1� k4�qsS2 ðskillÞDR ðA:13Þ

where DR is the repayment obligation (see the proof of Proposition 2). We know that rqXR < L. Thus, for1� k4 small enough, it follows by continuity that:

k4rqXR þ ½1� k4�qsS2 ðskillÞDR < L

in which case financiers liquidate all banks and a crisis ensues. h

References

Acharya, Viral, Naqvi, Hassan, 2012. The seeds of a crisis: a theory of bank liquidity and risk taking over the business cycle. J.Financ. Econ. 106 (2), 349–366.

Acharya, Viral, Thakor, Anjan V., 2014. The Dark Side of Liquidity Creation: Leverage-Induced Systemic Risk and Implications forthe Lender of Last Resort, Working Paper. NYU and Washington University in St. Louis.

Acharya, Viral V., Yorulmazer, Tanju, 2008. Cash-in-the-market pricing and the optimal resolution of bank failures. Rev. Financ.Stud. 21, 2705–2742.

Agarwal, Sumit, Benmelech, Efraim, Bergman, Nittai, Seru, Amit, 2012. Did the Community Reinvestment Act (CRA) Lead toRiskier Lending?, NBER, Working Paper #18609, December 2012.

Allen, Franklin, Babus, Ana, Carletti, Elena, 2012. Asset commonality, debt maturity, and systemic risk. J. Financ. Econ. 104 (3),519–534.

Allen, Franklin, Gale, Douglas, 2007. Understanding Financial Crises. Oxford University Press.Asea, Patrick, Blomberg, S. Brock, 2014. Lending Cycles, NBER Working Paper No. 5951, December 2014.Atkinson, Tyler, Luttrell, David, Rosenblum, Harvey, 2013. How Bad Was It? The Costs and Consequences of the 2007–2009

Financial Crisis. Staff Paper No. 20, Federal Reserve Bank of Dallas, July 2013.Bassett, William, Chosak, Mary Beth, Driscoll, John, Zakrajsek, Econ, 2014. Changes in bank lending standards and the

macroeconomy. J. Monet. Econ. 62, 23–40.

http://refhub.elsevier.com/S1042-9573(15)00030-3/h0005










Bekaert, Geert, Hoerova, Marie, Duca, Marco Lo, 2013. Risk, uncertainty and monetary policy. J. Monet. Econ. 60 (7), 771–788.Berger, Allen N., Bouwman, Christa H.S., 2013. How does capital affect bank performance during financial crises? J. Financ. Econ.,

146–176Bernanke, Ben, 1983. Nonmonetary effects of the financial crisis in the propagation of the great depression. Am. Econ. Rev. 73

(3), 257–276.Bernanke, Ben S., Gertler, Mark, 1995. Inside the black box: the credit channel of monetary policy transmission. J. Econ. Perspect.

9 (4), 27–48 (Fall).Boot, Arnoud W.A., Thakor, Anjan V., 1993. Self-interested bank regulation. Am. Econ. Rev. 83 (2), 206–212.Boot, Arnoud W.A., Thakor, Anjan V., 1994. Moral hazard and secured lending in an infinitely repeated credit market game. Int.

Econ. Rev. 35 (4), 899–920.Boot, Arnoud W.A., Thakor, Anjan V., 2000. Can relationship banking survive competition? J. Finance 55 (2), 679–713.Bordo, Michael, Eichengreen, Barry, Klingebiel, Daniela, Martinez-Peria, Maria Soledad, 2001. Is the crisis problem growing more

severe? Econ. Policy 16 (32), 53–82.Boyd, John, Kwak, Sungkyu, Smith, Bruce, 2005. The real output losses associated with modern banking crises. J. Money, Credit

Bank. 37, 977–999.Brunnermeier, Markus K., 2009. Deciphering the liquidity and credit crunch 2007–2008. J. Econ. Perspect. 23 (1), 77–100.Caballero, Ricardo, Simsek, Alp, 2013. Fire sales in a model of complexity. J. Finance 68 (6), 2549–2587.Calomiris, Charles W., Haber, Stephen H., 2014. Fragile by Design: The Political Origins of Banking Crises and Scarce Credit.

Princeton University Press.Calomiris, Charles W., Kahn, Charles M., 1991. The role of demandable debt in structuring optimal banking arrangements. Am.

Econ. Rev. 81 (3), 497–513.Chan, Yuk-Shee, Greenbaum, Stuart, Thakor, Anjan V., 1992. Is fairly-priced deposit insurance possible? J. Finance 47 (1), 227–

246.De Jonghe, Olivier, 2010. Back to the basics in banking? A micro-analysis of banking system stability. J. Financ. Intermed. 19 (3),

387–417.Dell’Ariccia, Giovanni, Igan, Deniz, Laeven, Luc, 2012. Credit booms and lending standards: evidence from the subprime

mortgage market. J. Money, Credit Bank. 44 (2–3), 284–367.Demirgiic-Kunt, Asli, Detragiache, Enrica, 1999. Does Deposit Insurance Increase Banking System Stability? An Empirical

Investigation, World Bank e-Library, November 1999.Farhi, Emanuel, Tirole, Jean, 2012. Collective moral hazard, maturity mismatch and systemic bailouts. Am. Econ. Rev. 102 (1),

60–93.Gennaioli, Nicola, Shleifer, Andrei, Vishny, Robert, 2012. Financial innovation and financial fragility. J. Financ. Econ. 104 (3),

452–468.Gennaioli, Nicola, Shleifer, Andrei, Vishny, Robert, 2015. Neglected risks: the psychology of financial crises. Am. Econ. Rev. 105

(5), 310–314.Gizycki, Marianne, 2001. The Effect of Macroeconomic Conditions on Banks’ Risk and Profitability. Research Discussion Paper

2001-06, Reserve Bank of Australia, September 2001.Goel, Anand, Song, Feghua, Thakor, Anjan, 2014. Correlated leverage and its ramifications. J. Financ. Intermed. 23 (4), 471–503.Gorton, Gary, 1988. Banking panics and the business cycle. Oxford Econ. Pap., New Series 40 (4), 751–781.Gorton, Gary, 2010. Slapped by the Invisible Hand: The Panic of 2007, vol. 6, No. 1, third ed. Oxford University Press, New York.Holmstrom, Bengt, Tirole, Jean, 1997. Financial intermediation, loanable funds, and the real sector. Quart. J. Econ. 112, 663–691.Jiménez, Gabriel, Ongena, Steven, Peydró, José-Luis, Saurina, Jesús, 2012. Credit supply and monetary policy: identifying the

bank balance-sheet channel with loan applications. Am. Econ. Rev. 102 (5), 2301–2326.Johnson, Simon, Kwak, James, 2010. 13 Bankers: The Wall Street Takeover and the Next Financial Meltdown. Random House,

Panther Books, New York.Kane, Edward, 2015. Regulation and supervision: an ethical perspective. In: Berger, A., Molyneux, P., Wilson, J. (Eds.), Oxford

Handbook on Banking, second ed. Oxford University Press, London (Chapter 12, forthcoming).Kane, Edward, 1990. Principal-agent problems in S&L salvage. J. Finance 45 (July), 755–764.Keys, Benjamin J., Mukherjee, Tanmoy, Seru, Amit, Vig, Vikrant, 2010. Did securitization lead to lax screening? Evidence from

subprime loans. Quart. J. Econ. 125 (1), 307–362.Kindleberger, Charles P., 1978. Manias Panics and Crashes: A History of Financial Crises. John Wiley & Sons, Inc..Krainer, John, 2004. What determines the credit spread? FRBSF Econ. Lett. (December 10)Kupiec, Paul, Ramirez, Carlos D., 2013. Bank failures and the cost of systemic risk: evidence from 1900–1930. J. Financ. Intermed.

22 (3), 285–307.Laeven, Luc, Valencia, Fabian, 2012. Systemic Banking Crises Database: An Update. IMF Working Paper WP/12/163,

International Monetary Fund, 2012.Lo, Andrew, 2012. Reading about the financial crisis: a 21-book review. J. Econ. Lit. 50 (1), 151–178.Maddaloni, Angela, Peydro’, Jose’-Luis, 2011. Bank risk-taking, securitization, supervision, and low interest rates: evidence from

the Euro-area and the U.S. lending standards. Rev. Financ. Stud. 24 (6), 2121–2165.Minsky, Hyman P., 1975. John Maynard Keynes. Columbia University Press, New York.Piketty, Thomas, 1995. Social mobility and redistributive policies. Quart. J. Econ. 110 (3), 551–584.Purnanandam, Amiyatosh, 2011. Originate-to-distribute model and the subprime mortgage crisis. Rev. Financ. Stud. 24 (6),

1881–1915.Rajan, Raghuram G., 2010. Fault Lines: How Hidden Fractures Still Threaten the World Economy. Princeton University Press,

Princeton and Oxford.Rajan, Raghuram G., 1994. Why bank credit policies fluctuate: a theory and some evidence. Quart. J. Econ. 109 (2), 399–441.Ramakrishnan, Ram T.S., Thakor, Anjan V., 1984. Information reliability and a theory of financial intermediation. Rev. Econ. Stud.

51 (3), 415–432.Reinhart, Carmen M., Rogoff, Kenneth, 2008. Is the 2007 U.S. sub-prime financial crisis so different? An international historical

comparison. Am. Econ. Rev. 98 (2), 339–344.
































































Rochet, Jean-Charles, 2008. Why are There So Many Banking Crises? The Politics and Policy of Bank Regulation. PrincetonUniversity Press.

Ruckes, M.E., 2004. Bank competition and credit standards studies. Rev. Financ. Stud. 17 (4), 1073–1102.Shleifer, Andrei, Vishny, Robert W., 2010. Unstable banking. J. Financ. Econ. 97, 306–318.Stiglitz, Joseph E., 2010. Free Fall. W. W. Norton and Company, New York.Taylor, John, B., 2009. The Financial Crisis and the Policy Responses: An Empirical Analysis of What Went Wrong. Working Paper

#14631, National Bureau of Economic Research (NBER), January 2009.Thakor, Anjan V., 2012. Incentives to innovate and financial crises. J. Financ. Econ. 103 (1), 130–148.Thakor, Anjan V., 2014. Bank capital and financial stability: economic tradeoff or faustian bargain? Annu. Rev. Financ. Econ. 6,

185–223.Thakor, Anjan V., 2015. Lending booms, smart bankers and financial crises. Am. Econ. Rev. 105 (5), 305–309.Thakor, Anjan V., 2015. The financial crisis of 2007–2009: why did it happen and what did we learn? Rev. Corp. Finance Stud.

(forthcoming).Wagner, Wolf, 2010. Diversification at financial intermediaries and systemic crises. J. Financ. Intermed. 19 (3), 373–386.Wilms, Philip, Swank, Job, de Haan, Jakob, 2014. Determinants of the Real Impact of Banking Crises: A Review and New

Evidence, Working Paper No. 437, De Nederlandsche Bank NV, August 2014.











j. finan. intermediation - olin business schoolapps.olin.wustl.edu › faculty › thakor ›...

Documents